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Integral Formula for Spectral Flow for $p$-Summable Operators

  • Magdalena Cecilia Georgescu,
    Department of Mathematics, Ben Gurion University, 8410501 Be'er Sheva, Israel
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Fix a von Neumann algebra $\mathcal{N}$ equipped with a suitable trace $\tau$. For a path of self-adjoint Breuer-Fredholm operators, the spectral flow measures the net amount of spectrum which moves from negative to non-negative. We consider specifically the case of paths of bounded perturbations of a fixed unbounded self-adjoint Breuer-Fredholm operator affiliated with $\mathcal{N}$. If the unbounded operator is p-summable (that is, its resolvents are contained in the ideal $L^p$), then it is possible to obtain an integral formula which calculates spectral flow. This integral formula was first proven by Carey and Phillips, building on earlier approaches of Phillips. Their proof was based on first obtaining a formula for the larger class of $\theta$-summable operators, and then using Laplace transforms to obtain a p-summable formula. In this paper, we present a direct proof of the p-summable formula, which is both shorter and simpler than theirs.
Keywords: spectral flow, $p$-summable Fredholm module spectral flow, $p$-summable Fredholm module
MSC Classifications: 19k56, 46L87, 58B34 show english descriptions unknown classification 19k56
Noncommutative differential geometry [See also 58B32, 58B34, 58J22]
Noncommutative geometry (a la Connes)
19k56 - unknown classification 19k56
46L87 - Noncommutative differential geometry [See also 58B32, 58B34, 58J22]
58B34 - Noncommutative geometry (a la Connes)

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